
where \(\hat{J}(s_1,s_2):=J( \psi (\theta _n(s_1)), n^{-1/2} s_2 )\) . Choose any \(\epsilon _2 > 0\) . By [1]}, the Jacobian term \(J\)  is smooth in \(s_2\)  for each \(s_1\) . WLOG, let \(\epsilon >0\)  be such that \(J\)  is \(\epsilon _2\)  close to 1. There must exist a \(\delta _{s_1}\)  such that
\(&\hspace{0.0pt}\leqslant \beta ^{-t/2} \int _{ S_n } \int _{||s_2|| \leqslant \delta _{s_1} } \exp \left( \frac{ s_2^{\prime } n \left( G^{\prime \prime }(\theta _n(s_1) ) + \epsilon _2 I\right)s_2 }{n\beta } \right) \left(\phi ^*_{0, I(\theta _0)}( s_1 ) + \epsilon _2 \right) (1 + \epsilon _2) ds_2 ~ ds_1 \\&\hspace{0.0pt} + \beta ^{-t/2} \int _{ S_n } \int _{ ||s_2||\in (\delta _{s_1}, n^{1/2} \epsilon ] } \exp \left( \frac{ nG( \theta _n(s_1) + n^{-1/2}s_2 Q_\theta ^\perp ) }{\beta } \right) \left(\phi ^*_{0, I(\theta _0)}( s_1 + s_2Q_\theta ^\perp ) \right) \hat{J}( s_1, s_2 ) ds_2 ~ ds_1 .\) 
